Problems in Combinatorics

56

Correct

That's right! We have to pick 3 objects out of 8 in the dataset. The order does not matter we cannot pick the same object twice. So we have combinations of size 3 out of 8 objects. There are 8x7x6/(3x2x1)=8x7=56 of them.

560

Correct

This is correct. When we pick K objects in the dataset of size N (K=3, N=8 for the first dataset and K=2, N=5 for the second), we are dealing with combinations: unordered selection without repetitions. Thus we can pick 3 objects in the first dataset in 8x7x6/(3x2x1)=8x7=56 ways and we can pick 2 objects from the second dataset in 5x4/(2x1)=10 ways. Since we want to pick a subset from both datasets, we are dealing with pairs of choices, so we have to multiply variants for both datasets.

7059052

Correct

This is correct! We have to pick a subset of size 6 out of 44 options. These are combinations and the answer is a binomial coefficient.

10545

Correct

This is correct! Since the lottery has already happened, the set of winning numbers is fixed. So to guess exactly four winning numbers one has to pick four numbers among 6 winning numbers and pick two other numbers among remaining 44-6=38 numbers. Choices for both cases can be computed as binomial coefficients and then we have to multiply the results by the product rule.

Also note that if we divide this number by the answer to the previous problem we get approximately 0.0015. As we will discuss next week this is the probability to guess exactly four numbers correctly, that is the chances are approximately 0.15%.

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